A 1D elastic plastic damage constitutive law for bone tissue

Motivated by applications in orthopaedic and maxillo-facial surgery, the mechanical behaviour of cortical bone in cyclic overloads at physiological strain rates is investigated. To this end, a new one-dimensional rate-independent constitutive model for compact bone is proposed to simulate the damage accumulation occurring during tensile or compressive overloading. We adopted a macroscopic and phenomenological description of the mechanics of cortical bone. The mathematical formulation of the model is established within the framework of generalized standard materials and is based on the definition of three internal state variables: a tensile and a compressive damage variable that represent the microcrack density and a residual strain variable that represents the permanent strain associated with the sliding behaviour of these microcracks. Distinct damage threshold stresses are used in tension and compression. As the macroscopic mechanical behaviour of bovine cortical bone is very similar to that of human cortical bone and of much easier access, we first achieved the validation and identification of the material constants of the constitutive laws in tension using new uniaxial experimental results of bovine compact bone of our own. With adequate original hardening rules, the constitutive model is able to reproduce the main features of cortical bone behaviour under arbitrary cyclic tensile and compressive loadings. The proposed algorithm applies for the first time three distinct projections based on the relationship between the three internal variables and criteria. The predicted stress-strain curves exhibit a damaged reloading which is collinear with the origin as many cyclic overloading experiments on cortical bone have shown. Note that as our model was identified for physiological strain rates, it hardly can be applied in high strain rate situations like the ones involved in impact studies.